On the Structure of the Linear Codes with a Given Automorphism

The purpose of this paper is to present the structure of the linear codes over a finite field with q elements that have a permutation automorphism of order m. These codes can be considered as generalized quasi-cyclic codes. Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail, presenting necessary and sufficient conditions for which linear codes with such an automorphism are self-orthogonal, self-dual, or linear complementary dual

Construction of isodual codes from polycirculant matrices

Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes. Self-duality, the trivial case of isoduality, can only occur over \F_2 in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over \F_2, we show that binary double polycirculant codes are asymptotically good

On Linear Codes over F2 x F2

A code of length n and size M consist of a set of M vectors of n components. The components being taken from some alphabet set S. So a code C is a set of n-tuples subset of Sn. If S has a ring structure then C is called a linear code over S if it is an S-module. To every linear code C there corresponds its dual C⊥, if C C⊥, then C is called self-orthogonal. If C = C⊥ then C is called self-dual. In this thesis we will study linear and self-dual codes over the rings of four alphabets and in more details over the ring F2 x F2, this ring is isomorphic to the ring F2 + vF2 where v2 = v and F2 = {0; 1}. We would also study linear and self-dual codes for other rings in the form Fp + vFp for different primes p. Also we will construct simplex code over the ring F2 + vF2≃ F2 x F2